Supremum

In analysis the supremum or least upper bound of a set S of real numbers is denoted by sup(S) and is defined to be the smallest real number that is greater than or equal to every number in S. If no such number exists (because S is not bounded above), then we define sup(S) = +∞. If S is empty, we define sup(S) = -∞ (see extended real number line).

An important property of the real numbers is that every set of real numbers has a supremum. This is sometimes called the supremum axiom and expresses the completeness of the real numbers.

Examples:

sup { x in R : 0 < x < 1 } = 1

sup { x in R : x² < 2 } = √2

sup { (-1)ⁿ - 1/n : n = 1, 2, 3, ...} = 1

Note that the supremum of S does not have to belong to S (like in these examples). If the supremum value belongs to the set then we can say there is a largest element in the set.

In general, in order to show that sup(S) ≤ A, one only has to show that x ≤ A for all x in S. Showing that sup(S) ≥ A is a bit harder: for any ε > 0, you have to exhibit an element x in S with x ≥ A - ε.

In functional analysis, one often considers the supremum norm of a bounded function f : X -> R (or C); it is defined as

||f||_∞ = sup { |f(x)| : x ∈ X }

and gives rise to several important Banach spaces.

See also: infimum or greatest lower bound, limit superior.

Generalization

One can define suprema for subsets S of arbitrary partially ordered sets (P, <=) as follows:

• A supremum or least upper bound of S is an element u in P such that
  • x <= u for all x in S, and
  • for any v in P such that x <= v for all x in S it holds that u <= v.

It can easily be shown that, if S has a supremum, then the supremum is unique: if u₁ and u₂ are both suprema of S then it follows that u₁ <= u₂ and u₂ <= u₁, and since <= is antisymmetric it follows that u₁ = u₂.

In an arbitrary partially ordered set, there may exist subsets which don't have a supremum. In a lattice every nonempty finite subset has a supremum, and in a complete lattice every subset has a supremum.